2019

Into the Unknown

A guide to estimating Type A measurement
uncertainty data

by Dilip Shah

For estimation of
measurement uncertainty,1-4 both
Type A and B uncertainty contributors may need to be considered, depending on
the parameter that is being estimated. ISO/IEC Guide
99:2007—International vocabulary of metrology—Basic and general
concepts and associated terms defines Type A evaluation of
measurement uncertainty as: "Evaluation of a component of measurement
uncertainty by a statistical analysis of measured quantity values obtained
under defined measurement conditions."5

One of the quickest ways of estimating Type
A measurement uncertainty data is by taking a series of measurements and
calculating its appropriate standard deviation. For example, if you took 10
measurements, as shown in Table 1, its standard deviation would be calculated
by using the sample standard deviation formula. This is also classified as
repeatability data in the measurement uncertainty budgets.

One of the reasons ISO/IEC 170256
accrediting bodies require accredited laboratories to estimate Type A
contributors to their measurement uncertainty budget is to ensure that the
laboratory is capable of making measurements with the equipment that they list
in their scope of accreditation. This validates the competence of the
laboratory and the personnel making the measurements.

The repeatability of measurements is
usually the minimum requirement for listing Type A uncertainty contribution to
the measurement uncertainty budget. However, any uncertainty contributor that
is estimated by statistical means should be listed as Type A data with its
appropriate degrees of freedom. Examples include:

When collecting Type A data for analysis,
ensure that the data collected takes into account the instrument resolution.7
If the instrument reads to a 0.001 resolution, ensure
that the data collected also reports to a 0.001 resolution. For example, a
0.500 reading on display is reported as 0.500. There should not be any
premature rounding of data. When data get collected by
automated data acquisition systems, ensure that it is capable of
collecting and reporting to the appropriate resolution.

In most cases, a laboratory has more than
one technician that is capable of making the measurements for that parameter.
Depending on the availability of the technician, either one may be assigned to
the task of making the measurement. In that case, it is important to know that
measurements made by either of the technicians are statistically insignificant
(reproducible) or if they are significant, the reproducibility of the
technicians need to be taken into account and listed as the second Type A
uncertainty contributor with its degrees of freedom.

Conducting such a study may require each of
the technicians to make 10 or more measurements on a single artifact, as shown
in Table 2, and employing appropriate statistical analysis. One-way analysis of
variance (ANOVA) is one technique that can be used. ANOVA is available in
popular spreadsheets and statistical software packages. It uses the
F-distribution to determine if there is a statistically significant difference
in the between groups (reproducibility) and within groups (repeatability) data.
It determines this by looking at the F-critical value (2.86626) from the
F-distribution table (degrees of freedom numerator of three and degrees of
freedom denominator of 36 at 95% confidence interval) and comparing it with the
F-calculated value, which is the between groups mean square (MS)-value divided
by within groups MS-value (see Online Figure 1 and Online Table 1 on this
column’s webpage at www.qualityprogress.com for more on the F-distribution). The MS-value is derived by dividing the sum of squares value by
degrees of freedom. It also is the variance value.

The decision rules are:

If
the F-value
is less than the F-critical value, then the data are considered
statistically insignificant. In this case, reproducibility may be ignored if it
does not make a significant overall contribution to measurement uncertainty.

If
the F-value
is greater than the F-critical value, then the data are considered
statistically significant. The reproducibility should be included in the
measurement uncertainty budget.

If
the F-value
is very close to the F-critical value, the data should be considered
statistically significant. The reproducibility should be included in the
measurement uncertainty budget. See Rule No. 4 for p-value (0.6195 = 61.95%).

Also
consider the p-value (0.6195 = 61.95%). It shows the probability
of significance. The higher it is, the better the confidence. This should also
be factored into the decision-making process.

To calculate reproducibility from the data
in Table 2, divide the square root of the MS-value for between groups by the
square root of 10 (or the appropriate sample size for the user’s data).

To calculate repeatability from the data in
Table 2, calculate the square root of the MS-value for within groups.

Ensure that the degrees of freedom
associated with Type A uncertainty are included in the measurement uncertainty
budget. Depending on the percent contribution to the overall uncertainty, the k
coverage factor may be dependent on the effective degrees of freedom using the
Welch Satterthwaite formula.8, 9

A future column will look at common Type B
uncertainty contributors, and how they are estimated for the measurement
uncertainty budget.

References

International Organization for
Standardization, ISO/IEC Guide
98-3:2008—Guide to the expression of uncertainty in measurement.

Dilip Shah is president of E = mc3 Solutions in Medina, OH.
He is chair of ASQ’s Measurement Quality Division and past chair of
Akron-Canton Section 0810. Shah is also co-author of The
Metrology Handbook (ASQ Quality Press, 2012). Shah is an ASQ-certified
quality engineer, quality auditor, calibration technician and an ASQ fellow.